Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

/pdf/levy-processes-and-stochastic-calculus-2rcj1efu2c.pdf

Lévy Processes and Stochastic Calculus

A method for simulating the Brownian dynamics of N particles with the inclusion of hydrodynamic interactions is described. The particles may also be subject to the usual interparticle or external forces (e.g., electrostatic) which have been included in previous methods for simulating Brownian dynamics of particles in the absence of hydrodynamic interactions. The present method is derived from the Langevin equations for the N particle assembly, and the results are shown to be consistent with the corresponding Fokker–Planck results. Sample calculations on small systems illustrate the importance of including hydrodynamic interactions in Brownian dynamics simulations. The method should be useful for simulation studies of diffusion limited reactions, polymer dynamics, protein folding, particle coagulation, and other phenomena in solution.

Brownian dynamics with hydrodynamic interactions

The spectrum of the Fokker-Planck operator for weakly coupled gases is considered. The operator is decomposed into operators acting on functions whose angular dependence is given by spherical harmonics. It is shown that the operator corresponding to l = 0 has zero for a point eigenvalue (the eigenfunction is the Maxwell distribution). There are no other point eigenvalues and the continuous spectrum of all of the operators is the entire negative real axis. Some consequences are briefly discussed.

On the theory of brownian motion

The effect of realistic boundary conditions on the computation of the specific heat of an isotropic solid at low temperatures is investigated. Two cases are considered: the surface free of stress and the surface rigidly clamped. The first of these is the one of physical interest. For both cases a term in the specific heat arises which is proportional to the surface area and to T2 and appreciably higher than Montroll's result in the case of the free solid. The effect of approximations introduced during the computation is estimated. Available experimental data are not adequate for a critical test of the theory.

Surface Specific Heat of an Isotropic Solid at Low Temperatures

A model for shape generation by strain and cell-cell adhesion in the epithelium of an arthropod leg segment

A series of equations are developed for the study of the effects of cosolvents on the solubility of a solute in mixed solutions where the solute displays a finite solubility. The equations differ depending on the scale used for the solute (and cosolvent) concentrations. The expressions use Kirkwood-Buff integrals to relate the changes in solubility to changes in the local solution composition around the solute and can be applied to study any type of ternary system including electrolyte cosolvents. The expressions provided here differ from previous approaches because of the use of a semi-open ensemble and the extension to finite solute solubilities.

/pdf/on-the-theory-of-solute-solubility-in-mixed-solvents-ujv1gwzwo3.pdf

On the theory of solute solubility in mixed solvents.

A theory is given for the ratio of initial slope to intercept in a plot of 1/D versus concentration of colloid particles; D is the self‐diffusion coefficient of the charged colloid particles (micelles). The theory starts from the Einstein relation, and the autocorrelation function expression for the friction constant. The main approximation is the use of Brownian motion theory to describe the effect of the solvent on micelle motion. There is good agreement with experiment in salt solutions, but the agreement is poor when there is no added salt. It is concluded that the theory is probably appropriate for solutions of high ionic strength, but the possibility that the data in salt‐free solution have not been interpreted properly, i.e., that the theory has a wider validity, is also discussed.

Robert M. Mazo

Papers

On the theory of brownian motion

Surface Specific Heat of an Isotropic Solid at Low Temperatures

A model for shape generation by strain and cell-cell adhesion in the epithelium of an arthropod leg segment

On the theory of solute solubility in mixed solvents.

On the Theory of the Concentration Dependence of the Self-Diffusion Coefficient of Micelles